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Truncation Dimension for Function Approximation

  • Peter KritzerEmail author
  • Friedrich Pillichshammer
  • Grzegorz W. Wasilkowski
Chapter

Abstract

We consider the approximation of functions of s variables, where s is very large or infinite, that belong to weighted anchored spaces. We study when such functions can be approximated by algorithms designed for functions with only very small number dimtrnc(ε, s) of variables. Here ε is the error demand and we refer to dimtrnc(ε, s) as the ε-truncation dimension. We show that for sufficiently fast decaying product weights and modest error demand (up to about ε ≈ 10−5) the truncation dimension is surprisingly very small.

Notes

Acknowledgements

The authors would like to thank two anonymous referees for their remarks that helped improving the presentation of the results in the paper.

P. Kritzer is supported by the Austrian Science Fund (FWF) Project F5506-N26 and F. Pillichshammer by the Austrian Science Fund (FWF) Project F5509-N26. Both projects are parts of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”.

References

  1. 1.
    Gnewuch, M., Hefter, M., Hinrichs, A., Ritter, K.: Embeddings of weighted Hilbert spaces and applications to multivariate and infinite-dimensional integration. J. Approx. Theory 222, 8–39 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Gnewuch, M., Hefter, M., Hinrichs, A., Ritter, K., Wasilkowski, G.W.: Equivalence of weighted anchored and ANOVA spaces of functions with mixed smoothness of order one in L p. J. Complex. 40, 78–99 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Hefter, M., Ritter, K.: On embeddings of weighted tensor product Hilbert spaces. J. Complex. 31, 405–423 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Hefter, M., Ritter, K., Wasilkowski, G.W.: On equivalence of weighted anchored and ANOVA spaces of functions with mixed smoothness of order one in L 1 and L norms. J. Complex. 32, 1–19 (2016)CrossRefGoogle Scholar
  5. 5.
    Hinrichs, A., Schneider, J.: Equivalence of anchored and ANOVA spaces via interpolation. J. Complex. 33, 190–198 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kritzer, P., Pillichshammer, F., Wasilkowski, G.W.: Very low truncation dimension for high dimensional integration under modest error demand. J. Complex. 35, 63–85 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kritzer, P., Pillichshammer, F., Wasilkowski, G.W.: On equivalence of anchored and ANOVA spaces; lower bounds. J. Complex. 38, 31–38 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kuo, F.Y., Sloan, I.H., Wasilkowski, G.W., Woźniakowski, H.: On decompositions of multivariate functions. Math. Comput. 79, 953–966 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kuo, F.Y., Schwab, C., Sloan, I.H.: Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal. 6, 3351–3374 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Sloan, I.H., Woźniakowski, H.: When are quasi-Monte Carlo algorithms efficient for high dimensional integrals? J. Complex. 14, 1–33 (1998)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Wasilkowski, G.W.: Tractability of approximation of -variate functions with bounded mixed partial derivatives. J. Complex. 30, 325–346 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Peter Kritzer
    • 1
    Email author
  • Friedrich Pillichshammer
    • 2
  • Grzegorz W. Wasilkowski
    • 3
  1. 1.Johann Radon Institute for Computational and Applied Mathematics (RICAM)Austrian Academy of SciencesLinzAustria
  2. 2.Department of Financial Mathematics and Applied Number TheoryJohannes Kepler University LinzLinzAustria
  3. 3.Computer Science DepartmentUniversity of KentuckyLexingtonUSA

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